(Op)lax natural transformations, twisted quantum field theories, and "even higher" Morita categories

TL;DR

This paper develops a comprehensive framework for lax and oplax transformations in higher categories, enabling the definition of twisted quantum field theories and extending Morita categories to higher dimensions.

Contribution

It introduces a double $( infty,n)$-category framework for (op)lax transformations and applies it to twisted quantum field theories and higher Morita categories.

Findings

Lax trivially-twisted relative field theories are equivalent to absolute theories.

Constructed a double $( infty,n)$-category for diagrammatics in higher categories.

Extended the higher Morita category of $E_d$-algebras to an $( infty,n+d)$-category.

Abstract

Motivated by the challenge of defining twisted quantum field theories in the context of higher categories, we develop a general framework for lax and oplax transformations and their higher analogs between strong $(\infty, n)$-functors. We construct a double $(\infty,n)$-category built out of the target $(\infty, n)$-category governing the desired diagrammatics. We define (op)lax transformations as functors into parts thereof, and an (op)lax twisted field theory to be a symmetric monoidal (op)lax natural transformation between field theories. We verify that lax trivially-twisted relative field theories are the same as absolute field theories. As a second application, we extend the higher Morita category of $E_d$-algebras in a symmetric monoidal $(\infty, n)$-category $\mathcal{C}$ to an $(\infty, n+d)$-category using the higher morphisms in $\mathcal{C}$.